# Krasner’s lemma

Krasner’s lemma (along with Hensel’s lemma) connects valuations^{} on fields to the algebraic structure^{} of the fields, and in particular to polynomial roots.

###### Lemma 1.

(Krasner’s Lemma) Let $K$ be a field of characteristic $\mathrm{0}$ complete^{} with respect to a nontrivial nonarchimedean absolute value^{}. Assume $\alpha \mathrm{,}\beta \mathrm{\in}\overline{K}$ (where $\overline{K}$ is some algebraic closure^{} of $K$) are such that for all nonidentity embeddings^{} $\sigma \mathrm{\in}{\mathrm{Hom}}_{K}\mathit{}\mathrm{(}K\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{,}\overline{K}\mathrm{)}$ we have $$. Then $K\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{\subset}K\mathit{}\mathrm{(}\beta \mathrm{)}$.

This says that for any $\alpha \in \overline{K}$, there is a neighborhood^{} of $\alpha $ each of whose elements generates at least the same field as $\alpha $ does.

###### Proof.

It suffices to show that for every $\sigma \in {\mathrm{Hom}}_{K(\beta )}(K(\alpha ,\beta ),\overline{K})$, we have $\sigma (\alpha )=\alpha $, for then $\alpha $ is in the fixed field of every embedding of $K(\beta )$, so $\alpha \in K(\beta )$. Note that

$$\left|\sigma (\alpha )-\beta \right|=\left|\sigma (\alpha )-\sigma (\beta )\right|=\left|\sigma (\alpha -\beta )\right|=\left|\alpha -\beta \right|$$ |

where the final equality follows since $\left|\sigma (\cdot )\right|$ is another absolute value extending ${|\cdot |}_{K}$ to $K(\alpha ,\beta )$ and thus must be equal to $|\cdot |$. But then

$$\left|\sigma (\alpha )-\alpha \right|=\left|(\sigma (\alpha )-\beta )+(\beta -\alpha )\right|\le \mathrm{max}(\left|\sigma (\alpha )-\beta \right|,\left|\alpha -\beta \right|)=\left|\alpha -\beta \right|$$ |

But this is impossible by the bounds on $\alpha ,\beta $ unless $\sigma (\alpha )=\alpha $. ∎

The first application of Krasner’s lemma is to show that splitting fields^{} are “locally constant” in the sense that sufficiently close polynomials^{} in $K[X]$ have the same splitting fields.

###### Proposition 2.

With $K$ as above, let $P\mathit{}\mathrm{(}X\mathrm{)}\mathrm{\in}K\mathit{}\mathrm{[}X\mathrm{]}$ be a monic irreducible polynomial^{} of degree $n$ with (distinct) roots ${\alpha}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\alpha}_{n}$. Then any monic polynomial^{} $Q\mathit{}\mathrm{(}X\mathrm{)}\mathrm{\in}K\mathit{}\mathrm{[}X\mathrm{]}$ of degree $n$ that is “sufficiently close” to $P\mathit{}\mathrm{(}X\mathrm{)}$ will be irreducible^{} over $K$ with roots ${\beta}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathit{}{\beta}_{n}$, and (after renumbering) $K\mathit{}\mathrm{(}{\alpha}_{i}\mathrm{)}\mathrm{=}K\mathit{}\mathrm{(}{\beta}_{i}\mathrm{)}$.

Here “sufficiently close” means the following: consider the space of degree $n$ polynomials over $K$ as homeomorphic to ${K}^{n}$ as a topological space^{}; close then means close in the obvious metric induced by $|\cdot |$.

###### Proof.

Since $P(X)$ has distinct roots, we may choose $$ for $i\ne j\le n$. Since the roots of a polynomial vary continuously with its coefficients, we say that a degree $n$ polynomial $Q(X)\in K[X]$ is sufficiently close to $P(X)$ if $Q(X)$ has roots ${\beta}_{1},\mathrm{\dots},{\beta}_{n}$ with $$. But ${\{{\alpha}_{j}\}}_{j\ne i}$ are all the Galois conjugates of ${\alpha}_{i}$, and $$ by construction, so by Krasner’s lemma, $K({\alpha}_{i})\subset K({\beta}_{i})$. But

$$[K({\beta}_{i}):K]\le \mathrm{deg}Q=\mathrm{deg}P=[K({\alpha}_{i}):K]$$ |

so that $K({\beta}_{i})=K({\alpha}_{i})$. In addition^{}, we see that $\mathrm{deg}Q=[K({\beta}_{i}):K]$ and thus that $Q(X)$ is irreducible.
∎

We use this fact to show that every finite extension^{} of ${\mathbb{Q}}_{p}$ arises as a completion of some number field^{}.

###### Corollary 3.

Let $K$ be a finite extension of ${\mathrm{Q}}_{p}$ of degree $n$. Then there is a number field $E$ and an absolute value $\mathrm{|}\mathrm{\cdot}\mathrm{|}$ on $E$ such that $\widehat{E}\mathrm{\cong}K$.

###### Proof.

Let $K={\mathbb{Q}}_{p}(\alpha )$ and let $P$ be the minimal polynomial for $\alpha $ over ${\mathbb{Q}}_{p}$. Since $\mathbb{Q}$ is dense in ${\mathbb{Q}}_{p}$, we can choose $Q(X)\in \mathbb{Q}[X]$ (note: in $\mathbb{Q}[X]$, not ${\mathbb{Q}}_{p}[X]$), and $\beta $ a root of $Q(X)$, as in the proposition^{}, so that ${\mathbb{Q}}_{p}(\alpha )={\mathbb{Q}}_{p}(\beta )$. Let $E=\mathbb{Q}(\beta )$. Clearly $E$ is a number field which, when regarded as embedded in ${\mathbb{Q}}_{p}(\beta )$, has absolute value ${|\cdot |}_{E}$, the restriction^{} of the absolute value on ${\mathbb{Q}}_{p}(\alpha )={\mathbb{Q}}_{p}(\beta )$. Then $\widehat{E}$ is a complete field with respect to that absolute value; ${\mathbb{Q}}_{p}(\beta )$ is as well, and $E$ is dense in both, so we must have $\widehat{E}={\mathbb{Q}}_{p}(\beta )={\mathbb{Q}}_{p}(\alpha )=K$.
∎

Finally, we can prove the following generalization^{} of Krasner’s Lemma, which is also given that name in the literature:

###### Lemma 4.

Let $K$ be a field of characteristic $\mathrm{0}$ complete with respect to a nontrivial nonarchimedean absolute value, and $\overline{K}$ an algebraic closure of $K$. Extend the absolute value on $K$ to $\overline{K}$; this extension^{} is unique. Let $\widehat{\overline{K}}$ be the completion of $\overline{K}$ with respect to this absolute value. Then $\widehat{\overline{K}}$ is algebraically closed.

###### Proof.

Let $\alpha $ be algebraic over $\widehat{\overline{K}}$ and $P(X)$ its monic irreducible polynomial in $\widehat{\overline{K}}[X]$. Since $\overline{K}$ is dense in $\widehat{\overline{K}}$, by proposition 2 we may choose $Q(x)\in \overline{K}[X]$ with a root $\beta \in \widehat{\overline{K}}$ such that $\widehat{\overline{K}}(\alpha )=\widehat{\overline{K}}(\beta )$. But $\widehat{\overline{K}}(\beta )=\widehat{\overline{K}}$ so that $\alpha \in \widehat{\overline{K}}$. ∎

Title | Krasner’s lemma |
---|---|

Canonical name | KrasnersLemma |

Date of creation | 2013-03-22 19:03:02 |

Last modified on | 2013-03-22 19:03:02 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 12J99 |

Classification | msc 11S99 |

Classification | msc 13H99 |